Mathematic root computer

ABSTRACT

Apparatus for electronically determining the mathematic root of a given value, comprising in series a plurality of integrators, a comparator, and a control circuit. A reference voltage is applied to the input of the first integrator and elicits an output from the last integrator, which output is compared by the comparator with an input voltage that reflects the value to be rooted. When the two compared voltages are equal, the control circuits undergoes a change of state, at which time the output voltage of the first integrator is a mathematic root of the given input value. The order of the root attained at the first integrator depends upon the number of integrators in the series; i.e., a series of three integrators will cause the output from the first integrator to be the cube root value of the input voltage. Moreover, other roots can be obtained from the output of the intermediate integrators.

United States Patent [151 3,679,884 Figueroa [451 July 25, 1972 54MATHEMATIC R001 .7 Primary Examiner-Eugene o. Botz Attorney-Silverman &Cass [72] lnventor: David R. Flgueroa, Dade County, Fla. [73] Assignee:Coulter Electronics, Inc., Hialeah, Fla. [57] ABSTRACT [22] Filed: Apr13 1970 Apparatus for electronically determining the mathematic root ofa given value, compnslng ll'l senes a plurality of integrators, [2|Appl. No.: 27,860 a comparator, and a control circuit. A referencevoltage is applied to the input of the first integrator and elicits anoutput from the last integrator, which output is compared by the 52 us.Cl ..235/193.5, 328/144 comparator with an input voltage that reflectsthe value to be [51] Int. Cl.. G06 7/20 rooted. When the two comparedvoltages are equal, the con- [58] Field ofSearch "235/1935, 183; 328/144trol circuits undergoes a h ng of State, at hi h im the output voltageof the first integrator is a mathematic root of the given input value.The order of the root attained at the first [56] References Citedintegrator depends upon the number of integrators in the se- UNITEDSTATES PATENTS rics; i.e., a series of three integrators will cause theoutput from the first integrator to be the cube root value of the input3,043,516 et al. X voltage Moreover other roots can be obtained from theout- Patchell X put of the intermediate integrators 3,521,046 7/1970Tippetts ..235/183 X 5 Claims, 1 Drawing Figure ii h 25 R 24 29 2 27 3|3 8 VREE E E E 26 30 32 I i l 33 RESET START T PATENIEQ 3,679,884

RESET START T Inventor DAVID R. FIGUEROA ATTYS.

MATHEMATIC ROOT COMPUTER BACKGROUND OF THE INVENTION The inventionrelates to electrical apparatus for determining the mathematic rootvalue of a given magnitude and, more particularly, the cube root valuethereof.

In many areas of technology it is desirable to have a fast, inexpensiveand proficient way to find the mathematic root of a given magnitude. Inthe field of computers, methods have been devised to find the squareroot of magnitudes. It is, however, believed that no methods have so farbeen developed for determining the cube root or roots of a higher orderof a given value.

Cube roots are of high practical significance in the fields of particlestudies. In such fields, dealing with particles of industrial, chemicalor biological nature, such as for instance in the study of bloodparticles, apparatus and methods have been developed for determiningparticle sizes in terms of their volumes. Assuming for example, thatsuch a particle is of the type of spheric body, and that it is ofinterest to know the diameter of such spheric body, a simple way offinding the diameter would be to cube root the numerical figurerepresenting the volume.

In other fields of science and even in that of pure mathematics it isoften desirable to find quickly and reliably roots of an order evenhigher than the third order such as the fourth, fifth order etc., aswell as the square root.

It is believed that electronic means so far have not been provided bythe prior art for a speedy determination of roots of an order exceedingsquare roots.

SUMMARY OF THE INVENTION Accordingly, it is an object of this inventionto overcome the deficiencies of the prior art by providing method andapparatus for the electronic determination of mathematic roots of anyorder.

It is a further object of the invention to provide a specificarrangement whereby the cube root of a given value is readilydetermined.

The invention is predicated on the idea that a reference voltage can beintegrated in an electronic integrator and elicit an output responsewhich increases with time. If such reference voltage is applied to aplurality of series connected integrators, the output response from eachsuccessive integrator will be a more steeply ascending curve than thatof the integrator feeding into it.

If there is a series of three, or four, or five, etc. integrators, thenby comparing the output of the last of the series with a given value,there will result at the output of the first integrator, respectively,the third, fourth, fifth, etc. root of the given value. Other rootvalues can be obtained from the output of the intermediate integrators.

A preferred embodiment is described for the case of cuberooting a givenvalue by using three series connected integrators, a comparator andcontrol circuit. A reference voltage is sent through the threeintegrators and the integrated output of the third integrator iscompared with a voltage constituting a value to be cube-rooted.Thereupon, such cube root is the output voltage of the first integrator.

BRIEF DESCRIPTION OF THE DRAWING The FIGURE is an electric schematic ofthe preferred embodiment.

DESCRIPTION OF THE PREFERRED EMBODIMENT Referring to the single figurewhich is a schematic illustration of the subject circuit and shows apreferred embodiment of the invention, it is readily noted that threeintegrators 21, 22 and 23 are series-connected. The first integrator 21comprises the series connection of a resistor R and a condenser C aswell as an amplifier 24 in parallel with the condenser C A switch 25 isconnected in series with the input to the first integrator 21 andhereinafter will be referred to as the input control switch 25. Also inparallel to the amplifier 24 is a switch 26 which will be referred to asthe parallel switch 26.

The second and third integrators 22 and 23 have elements and connectionscorresponding to those of the first integrator 21, specifically: aresistor R a condenser C and an amplifier 27; a resistor R a condenser Cand amplifier 28. Likewise, the integrator 22 is provided with an inputcontrol switch 29 and a parallel switch 30; whereas, the integrator 23is provided with an input control switch 31 and a parallel switch 32.

The output of the third integrator 23 is fed over a line 33 to one inputof a comparator 35. The output of the comparator 35 is coupled by a line36 to one input of control circuitry 37. A start input line 38 and areset input line 39 also are coupled to the control circuitry 37. Thelogic condition of these latter two input lines can beoperator-regulated as well as automatically regulated. Output from thecontrol circuitry is applied to a pair of lines 40 and 41, with the line40 determining the condition of the input control switches 25, 29 and31, and the output on the line 41 determining the condition of theparallel switches 26, 30 and 32.

A voltage summing junction 42 is at one input of the comparator 35 andis connected to a resistor 43 in the output line 33 from thirdintegrator 23. Also coupled to the summing junction is a resistor 44which receives its input from a source V which is the value to be cuberooted. It is to be assumed that the voltages at the resistors 43 and 44are of opposite polarity and thereby, when equal, provide a null voltageat the junction 42.

The other input 45 to the comparator is grounded. It will be appreciatedthat, in the alternative, the resistors 43 and 44 could be coupled tothe respectively different inputs to the comparator 35. In both modes ofcoupling, the attainment of specific voltages (typically equal voltages)at the resistors 43 and 44 will cause the comparator to produce acharacteristic output on the line 36 to activate the control circuitry37, as will be discussed next.

A reference voltage V is supplied to the first switch 25.

Other voltages of significance are:

E,, at the output of the first integrator 21;

E. at the output of the second integrator 22; and

E, at the output of the third integrator 23.

In operation, the reference voltage, preferably a fixed dc. voltage, isapplied to the first input control switch 25, which normally is open, asare the other input control switches 29 and 31; whereas, the parallelswitches 26, 30 and 32 normally are closed.

When the voltage V,,,, which is the quantum input representing amagnitude that is to be cube-rooted, is ready to be applied to the nulljunction 42, the start command is given on the line 38, whereby theinput control switches 25, 29 and 31 are closed, and simultaneously theparallel switches 26, 30 and 32 are opened, all at the same time. Theclosing of the input switches 25, 29 and 31 has the effect that allthree integrators 21, 22, 23 are charged by the value V and producevarying output responses, i.e. each shows an ascending output responsecurve representing a voltage as a function of time. More particularly,the output curve of the first integrator 21 is simply an ascendingstraight line or ramp voltage. The second integrator 22, since it isintegrating a ramp voltage, generates an output that is ascendingsomewhat faster than the straight line of the first curve, being in thenature of a function of the second power. The output response curve ofthe third integrator 23 has a still steeper ascending character, being afunction of the third power.

At the moment when E, the output voltage of the third integrator 23,reaches the level of V as seen at the null junction 42, the outputvoltage of the comparator 35 changes state and causes the input controlswitches 25, 29 and 31 to open simultaneously. At this moment of timeE,,, the output voltage of the first integrator 21, is:

-original state.

The above noted equation for E,, is derived in the folluwing way:

wherein k,#V,,,)/(R,C,),and r, equals the time when the comparatorchanges state. From (2) it follows that Since V,,,, R, and C, areconstants, k, necessarily is constant, and the curve for E, as afunction of t is an ascending straight line as noted heretofore.

The voltage at the output of the second integrator 22 iscorrespondingly:

E k2 a?) 4 wherein Finally, the voltage at the output of the thirdintegrator 23is:

V 8, emcmrczaaca (5) wherein k l 6RrCiR2C2 aca As stated heretofore, thechange of state by the comparator s which equation is identical with theequation i) set forth heretofore. y

Since I: is a pure number depending on pure numbers k, and

occurs when E V,,,, and therefore we may say:

' Since r,= (E,,)/(k,), as set forth above in equation (3), it followsthat:

( in m/( 1 Efl (Vi, 'k, )/(k) and therefore,

E i. 7 5 En kl k3 in or Eu: V3 in lfk is defined as 7 y k thcre can bederived E k V V,,,, 9)

wouldbei kvvm and the outputs from the second and third integrators 22and 23 would be:

Likewise, if there were six series connected integrators,

three integrators being added to those illustrated, and the outputfrom'the last being connected to the comparator:

Thus, it will be appreciated that any desired root value can beobtained, from the output from the first integrator 21 and, by properchoice of the number of integrators and the selection of outputs fromintermediate integrators, intermediate" root values obtained, such asquare root and cube root from the integrators 22 and 23.

It is believed that those skilled in the art will recognize and be ableto practice this invention from the foregoing and be able to adapt sameto particular use and in particular environments without departing fromthe spirit of the invention or the scope thereof as defined in thefollowing claims.

What is sought to be protected by United States Letters Patent is: i

1. Electronic apparatus for automatically producing at least onepredetermined root of an input quantum, said apparatus comprising:

a plurality of integrators connected in series,

each integrator having an input and an output,

an input control switch being series connected to the input of eachintegrator,

the input of the first of said plurality of integrators arranged toreceive a reference voltage,

the number of said integrators equaling the highest order of rootdirectly obtainable from an output from one of said integrators,

comparing means arranged to receive as inputs the input quantum and theoutput from the last of said plurality of integrators and to produce acharacteristic output when the magnitudes of these two inputs attain apredetermined relationship,

a control circuit coupled to receive the characteristic output from saidcomparing means,

said control circuit being arranged to be responsive to an 7 externallysupplied start command, and

said control circuit having a first output coupled to control theoperation of said input control switches in separate response to saidcharacteristic input from said comparing means and to a start command, I

whereupon, when said comparing means produces its characteristic output,the output from said first integrator demarcates said predetermined rootof the input quanturn.

2. Apparatus as set forth in claim 1 wherein the total k which in turndepend on constants V R, C,R,C, R C it is number of series connectedintegrators is three, and

obvious that k may easily be determined as a scale factor, and E, isreadily found as the cube-root of V,,,, which is the object of thisinvention.

Also within the teachings of this invention,,as will be appreciated fromthe foregoing is the fact that:

the comparing means is arranged to receive as an input the output fromthe third integrator,

whereby the output from the first integrator demarcates the cube root ofthe input quantum when the comparing means produces its characteristicoutput.

nally supplied reset command. and

said control circuit has a second output coupled to control theoperation of said reset switches in response to a reset command.

5. Apparatus as set forth in claim 2 wherein said comparing meanscomprises a comparator having an input summing junction coupled toreceive the output from the last integrator and the input quantum.

1. Electronic apparatus for automatically producing at least onepredetermined root of an input quantum, said apparatus comprising: aplurality of integrators connected in series, each integrator having aninput and an output, an input control switch being series connected tothe input of each integrator, the input of the first of said pluralityof integrators arranged to receive a reference voltage, the number ofsaid integrators equaling the highest order of root directly obtainablefrom an output from one of said integrators, comparing means arranged toreceive as inputs the input quantum and the output from the last of saidplurality of integrators and to produce a characteristic output when themagnitudes of these two inputs attain a predetermined relationship, acontrol circuit coupled to receive the characteristic output from saidcomparing means, said control circuit being arranged to be responsive toan externally supplied start command, and said control circuit having afirst output coupled to control the operation of said input controlswitches in separate response to said characteristic input from saidcomparing means and to a start command, whereupon, when said comparingmeans produces its characteristic output, the output from said firstintegrator demarcates said predetermined root of the input quantum. 2.Apparatus as set forth in claim 1 wherein the total number of seriesconnected integrators is three, and the comparing means is arranged toreceive as an input the output from the third integrator, whereby theoutput from the first integrator demarcates the cube root of the inputquantum when the comparing means produces its characteristic output. 3.Apparatus as set forth in claim 1 wherein second reset switch means areparallel coupled to each integrator switch, said control circuit isarranged to be responsive to an externally supplied reset command, andsaid control circuit has a second output coupled to control theoperation of said reset switches in response to a reset command. 4.Apparatus as set forth in claim 2 wherein second reset switch gate meansare parallel coupled to each integrator, said control circuit isarranged to be responsive to an externally supplied reset command, andsaid control circuIt has a second output coupled to control theoperation of said reset switches in response to a reset command. 5.Apparatus as set forth in claim 2 wherein said comparing means comprisesa comparator having an input summing junction coupled to receive theoutput from the last integrator and the input quantum.